In a low-scoring clash, Sammy steadied the ship for the tourists as they returning to winning ways in New Zealand following a comprehensive 2-0 Test series defeat earlier in the month.
The Black Caps looked to be in real trouble when they were skittled for just 156, despite the best efforts of skipper Brendon McCullum (51) and older brother Nathan (47).
The hosts bounced back with a five-wicket haul from Mitchell McClenaghan, removing Kieran Powell (5), Johnson Charles (9) and Darren Bravo (14) to leave West Indies stunned on 32-3. McLenaghan then also added the scalps of Dwayne Bravo and Danesh Ramdin to his tally.
But 157 is not the largest tally the WIndies will ever have to chase down and Sammy dug in to hit the winning runs, racking up his quick-fire 43 off 27 balls which included five fours and three sixes.
Earlier, Dwayne Bravo finished the NZ innings with figures of four for 44, with two wickets apiece for Ravi Rampaul - the key scalps of openers Martin Guptill and Jesse Ryder - and Jason Holder.
New Zealand's top four batsmen managed just 13 runs between them, Rampaul doing the early damage by dismissing Ryder and Guptill in his opening two overs.
Ryder, returning to international cricket after a 22-month absence, was caught at cover by Darren Bravo without having troubled the scorers, with Guptill (two) trapped lbw playing across the line.
They slumped to 10 for three in the seventh over when the in-form Ross Taylor (three) attempted a quick single and was caught well short by Holder.
Kane Williamson (eight) was next to go, his outside edge off the bowling of Holder snapped up by wicketkeeper Denesh Ramdin.
Dwayne Bravo then took centre stage with the ball. He snared the wicket of Corey Anderson (13) and then, after Sunil Narine had accounted for Luke Ronchi (seven), returned to see off James Neesham (10) and Brendon McCullum.
The 51-run knock from the skipper - who was dropped on 29 by Darren Sammy at point - came off just 57 balls, included five fours and a six and also saw him pass 5,000 runs in ODIs.
Nathan McCullum continued to dig in following his brother's departure, particularly in a last-wicket stand with Mitchell McClenaghan after Dwayne Bravo had had Kyle Mills (three) caught at first slip by Sammy.
The final partnership yielded 44 runs for the home team, but McCullum fell three runs short of his fifty when he was caught at backward square leg by Darren Bravo off the bowling of Holder.
McCullum had two fours and three sixes in his 64-ball 47, which went some way to ensuring the New Zealand bowlers had something to defend.